The state of a physical discrete-time system is assumed to satisfy
x
(
k
+
1
)
=
A
x
(
k
)
+
B
u
(
k
)
{\displaystyle \mathbf {x} (k+1)=A\mathbf {x} (k)+B\mathbf {u} (k)}
y
(
k
)
=
C
x
(
k
)
+
D
u
(
k
)
{\displaystyle \mathbf {y} (k)=C\mathbf {x} (k)+D\mathbf {u} (k)}
ملاحظ النمط المنزلق
x
^
˙
=
[
∂
H
(
x
^
)
∂
x
]
−
1
M
(
x
^
)
sgn
(
V
(
t
)
−
H
(
x
^
)
)
{\displaystyle \mathbf {\dot {\hat {x}}} =\left[{\frac {\partial H(\mathbf {\hat {x}} )}{\partial \mathbf {x} }}\right]^{-1}M(\mathbf {\hat {x}} )\,\operatorname {sgn} (V(t)-H(\mathbf {\hat {x}} ))}
where:
The
sgn
(
⋅
)
{\displaystyle \operatorname {sgn} ({\mathord {\cdot }})}
signum function to
n
{\displaystyle n}
sgn
(
z
)
=
[
sgn
(
z
1
)
sgn
(
z
2
)
⋮
sgn
(
z
i
)
⋮
sgn
(
z
n
)
]
{\displaystyle \operatorname {sgn} (\mathbf {z} )={\begin{bmatrix}\operatorname {sgn} (z_{1})\\\operatorname {sgn} (z_{2})\\\vdots \\\operatorname {sgn} (z_{i})\\\vdots \\\operatorname {sgn} (z_{n})\end{bmatrix}}}
for the vector
z
∈
R
n
{\displaystyle \mathbf {z} \in \mathbb {R} ^{n}}
The vector
H
(
x
)
{\displaystyle H(\mathbf {x} )}
h
(
x
)
{\displaystyle h(\mathbf {x} )}
H
(
x
)
≜
[
h
1
(
x
)
h
2
(
x
)
h
3
(
x
)
⋮
h
n
(
x
)
]
≜
[
h
(
x
)
L
f
h
(
x
)
L
f
2
h
(
x
)
⋮
L
f
n
−
1
h
(
x
)
]
{\displaystyle H(\mathbf {x} )\triangleq {\begin{bmatrix}h_{1}(\mathbf {x} )\\h_{2}(\mathbf {x} )\\h_{3}(\mathbf {x} )\\\vdots \\h_{n}(\mathbf {x} )\end{bmatrix}}\triangleq {\begin{bmatrix}h(\mathbf {x} )\\L_{f}h(\mathbf {x} )\\L_{f}^{2}h(\mathbf {x} )\\\vdots \\L_{f}^{n-1}h(\mathbf {x} )\end{bmatrix}}}
where
L
f
i
h
{\displaystyle L_{f}^{i}h}
i th Lie derivative of output function
h
{\displaystyle h}
f
{\displaystyle f}
x
{\displaystyle \mathbf {x} }
relative degree of n ,
H
(
x
(
t
)
)
{\displaystyle H(\mathbf {x} (t))}
y
(
t
)
=
h
(
x
(
t
)
)
{\displaystyle \mathbf {y} (t)=h(\mathbf {x} (t))}
n
−
1
{\displaystyle n-1}
Jacobian linearization of
H
(
x
)
{\displaystyle H(\mathbf {x} )}
H
(
x
)
{\displaystyle H(\mathbf {x} )}
diffeomorphism . The diagonal matrix
M
(
x
^
)
{\displaystyle M({\hat {\mathbf {x} }})}
M
(
x
^
)
≜
diag
(
m
1
(
x
^
)
,
m
2
(
x
^
)
,
…
,
m
n
(
x
^
)
)
=
[
m
1
(
x
^
)
m
2
(
x
^
)
⋱
m
i
(
x
^
)
⋱
m
n
(
x
^
)
]
{\displaystyle M({\hat {\mathbf {x} }})\triangleq \operatorname {diag} (m_{1}({\hat {\mathbf {x} }}),m_{2}({\hat {\mathbf {x} }}),\ldots ,m_{n}({\hat {\mathbf {x} }}))={\begin{bmatrix}m_{1}({\hat {\mathbf {x} }})&&&&&\\&m_{2}({\hat {\mathbf {x} }})&&&&\\&&\ddots &&&\\&&&m_{i}({\hat {\mathbf {x} }})&&\\&&&&\ddots &\\&&&&&m_{n}({\hat {\mathbf {x} }})\end{bmatrix}}}
where, for each
i
∈
{
1
,
2
,
…
,
n
}
{\displaystyle i\in \{1,2,\dots ,n\}}
m
i
(
x
^
)
>
0
{\displaystyle m_{i}({\hat {\mathbf {x} }})>0}
The observer vector
V
(
t
)
{\displaystyle V(t)}
V
(
t
)
≜
[
v
1
(
t
)
v
2
(
t
)
v
3
(
t
)
⋮
v
i
(
t
)
⋮
v
n
(
t
)
]
≜
[
y
(
t
)
{
m
1
(
x
^
)
sgn
(
v
1
(
t
)
−
h
1
(
x
^
(
t
)
)
)
}
eq
{
m
2
(
x
^
)
sgn
(
v
2
(
t
)
−
h
2
(
x
^
(
t
)
)
)
}
eq
⋮
{
m
i
−
1
(
x
^
)
sgn
(
v
i
−
1
(
t
)
−
h
i
−
1
(
x
^
(
t
)
)
)
}
eq
⋮
{
m
n
−
1
(
x
^
)
sgn
(
v
n
−
1
(
t
)
−
h
n
−
1
(
x
^
(
t
)
)
)
}
eq
]
{\displaystyle V(t)\triangleq {\begin{bmatrix}v_{1}(t)\\v_{2}(t)\\v_{3}(t)\\\vdots \\v_{i}(t)\\\vdots \\v_{n}(t)\end{bmatrix}}\triangleq {\begin{bmatrix}\mathbf {y} (t)\\\{m_{1}({\hat {\mathbf {x} }})\operatorname {sgn} (v_{1}(t)-h_{1}({\hat {\mathbf {x} }}(t)))\}_{\text{eq}}\\\{m_{2}({\hat {\mathbf {x} }})\operatorname {sgn} (v_{2}(t)-h_{2}({\hat {\mathbf {x} }}(t)))\}_{\text{eq}}\\\vdots \\\{m_{i-1}({\hat {\mathbf {x} }})\operatorname {sgn} (v_{i-1}(t)-h_{i-1}({\hat {\mathbf {x} }}(t)))\}_{\text{eq}}\\\vdots \\\{m_{n-1}({\hat {\mathbf {x} }})\operatorname {sgn} (v_{n-1}(t)-h_{n-1}({\hat {\mathbf {x} }}(t)))\}_{\text{eq}}\end{bmatrix}}}
where
sgn
(
⋅
)
{\displaystyle \operatorname {sgn} ({\mathord {\cdot }})}
signum function defined for scalars, and
{
…
}
eq
{\displaystyle \{\ldots \}_{\text{eq}}}
e
=
H
(
x
)
−
H
(
x
^
)
{\displaystyle \mathbf {e} =H(\mathbf {x} )-H(\mathbf {\hat {x}} )}
e
˙
=
d
d
t
H
(
x
)
−
d
d
t
H
(
x
^
)
=
d
d
t
H
(
x
)
−
M
(
x
^
)
sgn
(
V
(
t
)
−
H
(
x
^
(
t
)
)
)
,
{\displaystyle {\begin{aligned}{\dot {\mathbf {e} }}&={\frac {\operatorname {d} }{\operatorname {d} t}}H(\mathbf {x} )-{\frac {\operatorname {d} }{\operatorname {d} t}}H({\hat {\mathbf {x} }})\\&={\frac {\operatorname {d} }{\operatorname {d} t}}H(\mathbf {x} )-M({\hat {\mathbf {x} }})\,\operatorname {sgn} (V(t)-H({\hat {\mathbf {x} }}(t))),\end{aligned}}}
and so
[
e
˙
1
e
˙
2
⋮
e
˙
i
⋮
e
˙
n
−
1
e
˙
n
]
=
[
h
˙
1
(
x
)
h
˙
2
(
x
)
⋮
h
˙
i
(
x
)
⋮
h
˙
n
−
1
(
x
)
h
˙
n
(
x
)
]
⏞
d
d
t
H
(
x
)
−
M
(
x
^
)
sgn
(
V
(
t
)
−
H
(
x
^
(
t
)
)
)
⏞
d
d
t
H
(
x
^
)
=
[
h
2
(
x
)
h
3
(
x
)
⋮
h
i
+
1
(
x
)
⋮
h
n
(
x
)
L
f
n
h
(
x
)
]
−
[
m
1
sgn
(
v
1
(
t
)
−
h
1
(
x
^
(
t
)
)
)
m
2
sgn
(
v
2
(
t
)
−
h
2
(
x
^
(
t
)
)
)
⋮
m
i
sgn
(
v
i
(
t
)
−
h
i
(
x
^
(
t
)
)
)
⋮
m
n
−
1
sgn
(
v
n
−
1
(
t
)
−
h
n
−
1
(
x
^
(
t
)
)
)
m
n
sgn
(
v
n
(
t
)
−
h
n
(
x
^
(
t
)
)
)
]
=
[
h
2
(
x
)
−
m
1
(
x
^
)
sgn
(
v
1
(
t
)
⏞
v
1
(
t
)
=
y
(
t
)
=
h
1
(
x
)
−
h
1
(
x
^
(
t
)
)
⏞
e
1
)
h
3
(
x
)
−
m
2
(
x
^
)
sgn
(
v
2
(
t
)
−
h
2
(
x
^
(
t
)
)
)
⋮
h
i
+
1
(
x
)
−
m
i
(
x
^
)
sgn
(
v
i
(
t
)
−
h
i
(
x
^
(
t
)
)
)
⋮
h
n
(
x
)
−
m
n
−
1
(
x
^
)
sgn
(
v
n
−
1
(
t
)
−
h
n
−
1
(
x
^
(
t
)
)
)
L
f
n
h
(
x
)
−
m
n
(
x
^
)
sgn
(
v
n
(
t
)
−
h
n
(
x
^
(
t
)
)
)
]
.
{\displaystyle {\begin{aligned}{\begin{bmatrix}{\dot {\mathbf {e} }}_{1}\\{\dot {\mathbf {e} }}_{2}\\\vdots \\{\dot {\mathbf {e} }}_{i}\\\vdots \\{\dot {\mathbf {e} }}_{n-1}\\{\dot {\mathbf {e} }}_{n}\end{bmatrix}}&={\mathord {\overbrace {\begin{bmatrix}{\dot {h}}_{1}(\mathbf {x} )\\{\dot {h}}_{2}(\mathbf {x} )\\\vdots \\{\dot {h}}_{i}(\mathbf {x} )\\\vdots \\{\dot {h}}_{n-1}(\mathbf {x} )\\{\dot {h}}_{n}(\mathbf {x} )\end{bmatrix}} ^{{\tfrac {\operatorname {d} }{\operatorname {d} t}}H(\mathbf {x} )}}}-{\mathord {\overbrace {M({\hat {\mathbf {x} }})\,\operatorname {sgn} (V(t)-H({\hat {\mathbf {x} }}(t)))} ^{{\tfrac {\operatorname {d} }{\operatorname {d} t}}H(\mathbf {\hat {x}} )}}}={\begin{bmatrix}h_{2}(\mathbf {x} )\\h_{3}(\mathbf {x} )\\\vdots \\h_{i+1}(\mathbf {x} )\\\vdots \\h_{n}(\mathbf {x} )\\L_{f}^{n}h(\mathbf {x} )\end{bmatrix}}-{\begin{bmatrix}m_{1}\operatorname {sgn} (v_{1}(t)-h_{1}({\hat {\mathbf {x} }}(t)))\\m_{2}\operatorname {sgn} (v_{2}(t)-h_{2}({\hat {\mathbf {x} }}(t)))\\\vdots \\m_{i}\operatorname {sgn} (v_{i}(t)-h_{i}({\hat {\mathbf {x} }}(t)))\\\vdots \\m_{n-1}\operatorname {sgn} (v_{n-1}(t)-h_{n-1}({\hat {\mathbf {x} }}(t)))\\m_{n}\operatorname {sgn} (v_{n}(t)-h_{n}({\hat {\mathbf {x} }}(t)))\end{bmatrix}}\\&={\begin{bmatrix}h_{2}(\mathbf {x} )-m_{1}({\hat {\mathbf {x} }})\operatorname {sgn} ({\mathord {\overbrace {{\mathord {\overbrace {v_{1}(t)} ^{v_{1}(t)=y(t)=h_{1}(\mathbf {x} )}}}-h_{1}({\hat {\mathbf {x} }}(t))} ^{\mathbf {e} _{1}}}})\\h_{3}(\mathbf {x} )-m_{2}({\hat {\mathbf {x} }})\operatorname {sgn} (v_{2}(t)-h_{2}({\hat {\mathbf {x} }}(t)))\\\vdots \\h_{i+1}(\mathbf {x} )-m_{i}({\hat {\mathbf {x} }})\operatorname {sgn} (v_{i}(t)-h_{i}({\hat {\mathbf {x} }}(t)))\\\vdots \\h_{n}(\mathbf {x} )-m_{n-1}({\hat {\mathbf {x} }})\operatorname {sgn} (v_{n-1}(t)-h_{n-1}({\hat {\mathbf {x} }}(t)))\\L_{f}^{n}h(\mathbf {x} )-m_{n}({\hat {\mathbf {x} }})\operatorname {sgn} (v_{n}(t)-h_{n}({\hat {\mathbf {x} }}(t)))\end{bmatrix}}.\end{aligned}}}
وبذلك:
ما دام
m
1
(
x
^
)
≥
|
h
2
(
x
(
t
)
)
|
{\displaystyle m_{1}(\mathbf {\hat {x}} )\geq |h_{2}(\mathbf {x} (t))|}
e
˙
1
=
h
2
(
x
^
)
−
m
1
(
x
^
)
sgn
(
e
1
)
{\displaystyle {\dot {\mathbf {e} }}_{1}=h_{2}({\hat {\mathbf {x} }})-m_{1}({\hat {\mathbf {x} }})\operatorname {sgn} (\mathbf {e} _{1})}
e
1
=
0
{\displaystyle e_{1}=0}
Along the
e
1
=
0
{\displaystyle e_{1}=0}
v
2
(
t
)
=
{
m
1
(
x
^
)
sgn
(
e
1
)
}
eq
{\displaystyle v_{2}(t)=\{m_{1}({\hat {\mathbf {x} }})\operatorname {sgn} (\mathbf {e} _{1})\}_{\text{eq}}}
h
2
(
x
)
{\displaystyle h_{2}(\mathbf {x} )}
v
2
(
t
)
−
h
2
(
x
^
)
=
h
2
(
x
)
−
h
2
(
x
^
)
=
e
2
{\displaystyle v_{2}(t)-h_{2}({\hat {\mathbf {x} }})=h_{2}(\mathbf {x} )-h_{2}({\hat {\mathbf {x} }})=\mathbf {e} _{2}}
m
2
(
x
^
)
≥
|
h
3
(
x
(
t
)
)
|
{\displaystyle m_{2}(\mathbf {\hat {x}} )\geq |h_{3}(\mathbf {x} (t))|}
e
˙
2
=
h
3
(
x
^
)
−
m
2
(
x
^
)
sgn
(
e
2
)
{\displaystyle {\dot {\mathbf {e} }}_{2}=h_{3}({\hat {\mathbf {x} }})-m_{2}({\hat {\mathbf {x} }})\operatorname {sgn} (\mathbf {e} _{2})}
e
2
=
0
{\displaystyle e_{2}=0}
Along the
e
i
=
0
{\displaystyle e_{i}=0}
v
i
+
1
(
t
)
=
{
…
}
eq
{\displaystyle v_{i+1}(t)=\{\ldots \}_{\text{eq}}}
h
i
+
1
(
x
)
{\displaystyle h_{i+1}(\mathbf {x} )}
m
i
+
1
(
x
^
)
≥
|
h
i
+
2
(
x
(
t
)
)
|
{\displaystyle m_{i+1}(\mathbf {\hat {x}} )\geq |h_{i+2}(\mathbf {x} (t))|}
(
i
+
1
)
{\displaystyle (i+1)}
th row of the error dynamics,
e
˙
i
+
1
=
h
i
+
2
(
x
^
)
−
m
i
+
1
(
x
^
)
sgn
(
e
i
+
1
)
{\displaystyle {\dot {\mathbf {e} }}_{i+1}=h_{i+2}({\hat {\mathbf {x} }})-m_{i+1}({\hat {\mathbf {x} }})\operatorname {sgn} (\mathbf {e} _{i+1})}
e
i
+
1
=
0
{\displaystyle e_{i+1}=0}
So, for sufficiently large
m
i
{\displaystyle m_{i}}
m
i
{\displaystyle m_{i}}
|
h
i
(
x
(
0
)
)
|
{\displaystyle |h_{i}(\mathbf {x} (0))|}
H
:
R
n
→
R
n
{\displaystyle H:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}
diffeomorphism (i.e., that its Jacobian linearization is invertible) asserts that convergence of the estimated output implies convergence of the estimated state. That is, the requirement is an observability condition.
In the case of the sliding mode observer for the system with the input, additional conditions are needed for the observation error to be independent of the input. For example, that
∂
H
(
x
)
∂
x
B
(
x
)
{\displaystyle {\frac {\partial H(\mathbf {x} )}{\partial \mathbf {x} }}B(\mathbf {x} )}
does not depend on time. The observer is then
x
^
˙
=
[
∂
H
(
x
^
)
∂
x
]
−
1
M
(
x
^
)
sgn
(
V
(
t
)
−
H
(
x
^
)
)
+
B
(
x
^
)
u
.
{\displaystyle {\dot {\mathbf {\hat {x}} }}=\left[{\frac {\partial H(\mathbf {\hat {x}} )}{\partial \mathbf {x} }}\right]^{-1}M(\mathbf {\hat {x}} )\operatorname {sgn} (V(t)-H(\mathbf {\hat {x}} ))+B(\mathbf {\hat {x}} )u.}
![{\displaystyle \mathbf {\dot {\hat {x}}} =\left[{\frac {\partial H(\mathbf {\hat {x}} )}{\partial \mathbf {x} }}\right]^{-1}M(\mathbf {\hat {x}} )\,\operatorname {sgn} (V(t)-H(\mathbf {\hat {x}} ))}](https://www.marefa.org/api/rest_v1/media/math/render/svg/f6d22465e51d688658daa689d34c89ab34b824fc)
![{\displaystyle {\dot {\mathbf {\hat {x}} }}=\left[{\frac {\partial H(\mathbf {\hat {x}} )}{\partial \mathbf {x} }}\right]^{-1}M(\mathbf {\hat {x}} )\operatorname {sgn} (V(t)-H(\mathbf {\hat {x}} ))+B(\mathbf {\hat {x}} )u.}](https://www.marefa.org/api/rest_v1/media/math/render/svg/52351e91aac520f6330a9662a749215ffcd6d9a5)
